In the following exercise, find the smallest interval for theta [from this point on i will denote theta as '*' for ease of typing] starting with * min= 0 so that your graphing utility graphs the given polar equation exactly once without retracing any portion of it.
r= 4sin(*)
I'm using a graphing calculator to do this (TI 83-Plus) but I'm having some trouble with it. Is there a method to doing this or is it a guess-and-check number punching thing? I'm having a hard time determining when my graph is repeating itself and when it isn't. Any help would be appreciated.
To find the smallest interval for theta (*), where the graph of the polar equation r = 4sin(*) is plotted exactly once without retracing any portion, we need to consider the period of the function sin(*).
The function sin(*) has a period of 2*pi, which means it repeats every 2*pi radians. In other words, the graph of sin(*) will start repeating itself after every 2*pi radians.
In the given equation r = 4sin(*), when we plot the graph, we want to make sure that the entire graph is covered within one full period of sin(*). This will ensure that the polar equation is graphed exactly once without any repetition.
To determine the interval for theta (*), we can set up the following inequality:
0 <= * < 2*pi
Here, we start at * = 0 (as given in the problem) and go up to but not including * = 2*pi. This interval guarantees that we will cover one full period of sin(*) without any repetition.
Now, let's use your TI-83 Plus graphing calculator to plot the graph and verify our interval for theta.
1. Turn on the calculator and press the "MODE" button.
2. Set the mode to "RADIAN" to work with radians.
3. Press the "Y=" button to enter the equation.
4. Enter "4sin(T)" in the first equation slot (without quotes). Remember to use "T" instead of "*".
5. Press the "ZOOM" button and select option 6 for "ZStandard" to set the default zoom level.
6. Press the "GRAPH" button to plot the graph.
The resulting graph should show the complete graph of r = 4sin(*) within the interval defined as 0 <= * < 2*pi. If you see any repeated portions or extra segments beyond this interval, you may need to adjust the window settings or the zoom level to focus on the desired interval.
Remember, when working with polar equations, it's important to set the appropriate window and zoom settings to visualize the graph accurately.
I hope this helps you plot the graph of the given polar equation exactly once without retracing any portion. Let me know if you need any further assistance!